arXiv subjects

Enea G. Bongiorno

Publications and source records attributed to Enea G. Bongiorno.

5 recordsLinked to original sources

arXiv2026

Principal Component Analysis for Zero-Inflated Probability Measures via Lorenz Curves

This paper develops a principal component analysis framework for probability measures associated with non-negative real-valued random variables with structural zeros. Specifically, leveraging the associated Lorenz Curve (LC) and a suitable perspective transformation of the plane, a bijective embedding is defined to treat such measures as functional data in a Hilbert space. Once the random LC is embedded in this space, standard Functional Principal Component Analysis (FPCA) becomes directly applicable, and the diffeomorphic nature of the embedding allows the main convergence results and their interpretations to be transferred back to the original space of LC. The a.s. and $L^2(\mathbb{P})$ consistency of the estimated eigenfunctions, eigenvalues, and modes of variation is subsequently investigated, both in the embedded space and, by virtue of the embedding's diffeomorphism property, in the original LC space. Crucially, this consistency analysis accounts for the double stochasticity of the estimation problem: one level of randomness is tied to the underlying LC process itself, while the other stems from the empirical estimation of the LC from sample data. As a consequence, consistency of the Karhunen--Loève reconstructions is also established, guaranteeing faithful approximation of individual curves in both spaces. Finally, the proposed FPCA methodology is illustrated on a real-world dataset of global export flows and compared with a naive FPCA approach.

stat.ME↗

arXiv2016

Classification methods for Hilbert data based on surrogate density

An unsupervised and a supervised classification approaches for Hilbert random curves are studied. Both rest on the use of a surrogate of the probability density which is defined, in a distribution-free mixture context, from an asymptotic factorization of the small-ball probability. That surrogate density is estimated by a kernel approach from the principal components of the data. The focus is on the illustration of the classification algorithms and the computational implications, with particular attention to the tuning of the parameters involved. Some asymptotic results are sketched. Applications on simulated and real datasets show how the proposed methods work.

stat.ME↗

arXiv2011

A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let $X$ be a fuzzy set--valued random variable (\frv{}), and $\huku{X}$ the family of all fuzzy sets $B$ for which the Hukuhara difference $X\HukuDiff B$ exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be decomposed as $X(ω)=C\Mink Y(ω)$ for $\mathbb{P}$--almost every $ω\inΩ$, $C$ is the unique deterministic fuzzy set that minimizes $\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in $\huku{X}$, and $Y$ is a centered \frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \frv{} translation (i.e. $X(ω) = M \Mink \indicator{ξ(ω)}$ for some deterministic fuzzy convex set $M$ and some random element in $\Banach$). In particular, $X$ is an \frv{} translation if and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a translation. Examples, such as the Gaussian case, are provided.

math.PR↗

arXiv2008

Statistical aspects of birth--and--growth stochastic processes

The paper considers a particular family of set--valued stochastic processes modeling birth--and--growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set--valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.

stat.AP↗

arXiv2008

A set-valued framework for birth-and-growth process

We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point.

math.PR↗